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Rational Method Calculator (Q = CiA) — Weighted C, With Exam Examples
Estimate peak runoff from a small watershed with the Rational Method — single surface or an area-weighted multi-surface builder, with frequency factors applied correctly (and capped). Pre-loaded with a real exam-style problem.
Switching converts every value into the new system (it doesn't just relabel). Your choice is remembered on this device.
For study and checking only. These calculators run entirely in your browser — nothing is sent anywhere. They are meant to verify your hand calculations while you study; on the exam you will work by hand with the NCEES reference handbook.
Quick answer: Q = CiA
US units: Q [cfs] = 1.008333 · Cf · C · i [in/hr] · A [ac] | SI units: Q [m³/s] = Cf · C · i [mm/hr] · A [ha] / 360
The 1.008333 factor is the exact unit conversion (1 ac·in/hr = 1.008333 cfs). Many textbooks and some exam solutions use the approximation Q ≈ CiA (factor 1.0) — a 0.8% difference, inside answer-choice spacing on most problems, but know which one your reference uses. The calculator shows both.
Why area-weighting (never a simple average)
For mixed land use, the composite coefficient is Cw = Σ(Ci·Ai) / ΣAi — each surface's C weighted by its own area. A simple average of the C values is wrong unless every surface has the same area. The weighted builder above does this for you and shows each surface's share of the total runoff — notice how a small paved area often dominates the answer.
The effective coefficient is Ceff = min(Cf·C, 1.0): runoff can't exceed rainfall, so the frequency-factor product is capped at 1.0 and the calculator says so when the cap bites.
Worked example 1 (PE pace): weighted C with frequency factor
A 5.0-ac site has 2.5 ac of lawn (C = 0.20), 1.5 ac of pavement (C = 0.95), and 1.0 ac of roofs (C = 0.90). The 25-year IDF intensity at t = tc is 4.2 in/hr. Find the peak runoff. Flip the calculator to "Weighted" above — these rows are pre-loaded.
- A = 2.5 + 1.5 + 1.0 = 5.0 ac
- Cw = (2.5×0.20 + 1.5×0.95 + 1.0×0.90)/5.0 = (0.50 + 1.425 + 0.90)/5.0 = 2.825/5.0 = 0.565
- 25-yr → Cf = 1.1. Cf·Cw = 1.1 × 0.565 = 0.6215 — under the 1.0 cap, so Ceff = 0.6215
- Q = 1.008333 × 1.1 × 0.565 × 4.2 × 5.0 = 13.16 cfs (= 5907 gpm)
Runoff shares: the 1.5 ac of pavement (30% of the area) produces 50.4% of the runoff — the teaching point of area-weighting. The ≈1.0-factor approximation gives 13.05 cfs, 0.8% lower.
Worked example 2 (FE pace): single surface
A 10-ac parking lot (C = 0.90) sees 5.0 in/hr at the design return period (Cf = 1.0). Find Qp.
- Q = 1.008333 × 1.0 × 0.90 × 5.0 × 10 = 45.38 cfs (= 20366 gpm = 70.21 MGD)
- The ≈1.0 approximation: 0.90 × 5.0 × 10 = 45.00 cfs.
This is the calculator's default load — it matches on first paint.
Worked example 3: single surface (SI)
A 2.0 ha paved catchment (C = 0.75) is designed for a storm intensity of 100 mm/h. Find the peak runoff.
- Q = C·i·A/360 = (0.75 × 100 × 2.0)/360 = 0.417 m³/s
Exam tip: the 360 divisor is the unit conversion for i in mm/h and A in hectares. In US units (i in in/hr, A in acres) Q ≈ C·i·A directly. The rational method is a small-catchment method — question it for areas above ~80 ha.
Free formula resources
Grab the free formula resources — every FE Civil equation in one searchable index, with the Rational Method in both unit systems and the frequency-factor table.
Runoff coefficients — reference table
Ranges on an HEC-22 basis (surface × slope × return period style). The calculator's presets insert the range midpoint, labelled as such.
| Surface | C range |
|---|---|
| Asphalt / concrete pavement | 0.70–0.95 |
| Roofs | 0.75–0.95 |
| Lawns, sandy soil (flat / average / steep) | 0.05–0.20 |
| Lawns, heavy soil (flat / average / steep) | 0.13–0.35 |
| Gravel | 0.15–0.30 |
| Undeveloped / wooded | 0.10–0.30 |
| Downtown business | 0.70–0.95 |
| Residential, single-family | 0.30–0.50 |
Frequency factors
| Return period | Cf |
|---|---|
| 2–10 yr | 1.0 |
| 25 yr | 1.1 |
| 50 yr | 1.2 |
| 100 yr | 1.25 |
Cf scales the design C for rarer storms (a rarer storm saturates the watershed, so more of it runs off). The product Cf·C is capped at 1.0 — applied to the (weighted) coefficient, not to any single surface alone when Cf = 1.0.
The tc dependency: intensity comes from IDF at t = tc
The intensity i is not "the storm's intensity" in the abstract — it is the IDF-curve intensity at a duration equal to the time of concentration, the travel time from the hydraulically most distant point to the outlet. Shorter tc → higher intensity → larger Q. The method assumes the whole watershed contributes, which is only true when the storm duration is at least tc.
Exam problems usually hand you the IDF value (as in the worked examples). In practice you compute tc first — a time-of-concentration calculator is on the way; meanwhile the method is covered in Topic 7: Hydrology & Runoff and drilled in the hydrology practice problems.
Common exam traps
- Unweighted C. Averaging the C values instead of area-weighting them. With unequal areas this is always wrong — and always one of the answer choices.
- Wrong return-period column. Reading the intensity from the 10-yr IDF column when the problem asks for the 25-yr design — or applying Cf to an intensity that already came from the design storm's IDF curve.
- Area-unit errors. Plugging ft² in as acres (off by 43,560×) or m² in as hectares (off by 10,000×). The calculator's field labels name the expected unit; the exam won't.
- Forgetting the Cf cap. Cf·C = 1.1875 doesn't mean 118.75% of the rain runs off — it means Ceff = 1.0. Worked example: C = 0.95, i = 6.0 in/hr, A = 5 ac, 100-yr (Cf = 1.25) → Q = 1.008333 × 1.0 × 6.0 × 5.0 = 30.25 cfs, with the cap note shown.
- Rational Method on a 500-acre watershed. It's a small-watershed method — most references cap it near 200 ac. Above that the exam wants the Curve Number method or a unit hydrograph (the hydrology topic page covers the Curve Number method; a dedicated calculator is on its way).
Rational Method — FAQ
Is the factor 1.0 or 1.008?
Both. The exact conversion of 1 ac·in/hr to cfs is 1.008333 (43560/43200); many textbooks round to 1.0 and write Q ≈ CiA. The difference is 0.8% — inside answer-choice spacing on most problems, but if two choices straddle it, use the reference your exam allows. The calculator shows both.
When do I use weighted C?
Whenever the watershed has more than one surface type. Compute Cw = Σ(CiAi)/ΣAi — area-weighted, never a simple average of the C values.
What if C·Cf > 1?
Cap it at 1.0. Runoff can't exceed rainfall, so Ceff = min(Cf·C, 1.0). The calculator applies the cap and tells you when it does.
Rational Method or Curve Number — which one?
Rational for small, fairly uniform watersheds (up to ~200 ac) when you need a peak rate and have an IDF intensity at t = tc. Curve Number when you need a runoff depth/volume from a design-storm depth, or the watershed is larger and mixed. The exam tests knowing which method fits the given data.
What tc do I use for the intensity?
The time of concentration of the watershed — travel time from the hydraulically most distant point. Read the IDF curve at duration = tc. Using the wrong duration is the #1 Rational Method error in practice and on exams.
Keep building exam speed
Hydrology is the largest area on the PE Civil Water Resources & Environmental exam — 8–12 of the 80 questions. The PE WRE Flagship ($119) drills every one of those question types — soft launch, join the waitlist from the contact page. For a full rehearsal under timed conditions, the 110-question FE Civil exam simulation is a 5-hour-20-minute run with detailed solutions.
Last reviewed: 2026-10-03. Formulas follow the NCEES FE Reference Handbook conventions; always confirm against the current handbook.